Solve the equation $x^4-8x^3+23x^2-30x+15=0$

Question

Solve the equation $$x^4-8x^3+23x^2-30x+15=0$$ As $x=0$ is obviously not a solution, we can consider $x\ne0$, so I have tried to divide both sides by $x^2$ to get $$x^2-8x+23-\dfrac{30}{x}+\dfrac{15}{x^2}=0$$ Clearly this does not help. I have also tried to find the rational roots using Horner's method. It seems that the polynomial in the left-hand side does not have rational roots.

Answer 1

Note that, if $p(x)$ is your polynomial, then $p(x+2)=x^4-x^2-2x-1=x^4-(x+1)^2$. Can you take it from here?

Answer 2

Since you tried the rational root theorem and didn't find any rational roots, then next thing to try is factoring as a product of two quadratics. Set your polynomial equal to

$$(x^2+ax+b)(x^2+cx+d)$$ $$=x^4 + (a+c)x^3+(d+ac+b)x^2+(ad+bc)x+bd$$

and equate coefficients. Since $bd=15$ you can try $b=3$, $d=5$ and see what $a$ and $c$ turn out to be. If that doesn't work, try $b=1$, $d=15$.